| L(s) = 1 | + 21.2·2-s + 321.·4-s + 386.·5-s + 166.·7-s + 4.10e3·8-s + 8.20e3·10-s − 4.09e3·11-s + 1.18e4·13-s + 3.53e3·14-s + 4.58e4·16-s + 6.00e3·17-s − 3.62e4·19-s + 1.24e5·20-s − 8.67e4·22-s + 3.15e4·23-s + 7.14e4·25-s + 2.50e5·26-s + 5.36e4·28-s + 646.·29-s + 1.14e5·31-s + 4.47e5·32-s + 1.27e5·34-s + 6.45e4·35-s + 3.90e5·37-s − 7.69e5·38-s + 1.58e6·40-s + 4.13e5·41-s + ⋯ |
| L(s) = 1 | + 1.87·2-s + 2.51·4-s + 1.38·5-s + 0.183·7-s + 2.83·8-s + 2.59·10-s − 0.926·11-s + 1.49·13-s + 0.344·14-s + 2.80·16-s + 0.296·17-s − 1.21·19-s + 3.47·20-s − 1.73·22-s + 0.541·23-s + 0.914·25-s + 2.79·26-s + 0.461·28-s + 0.00492·29-s + 0.687·31-s + 2.41·32-s + 0.555·34-s + 0.254·35-s + 1.26·37-s − 2.27·38-s + 3.92·40-s + 0.936·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(12.19419865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.19419865\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 + 2.05e5T \) |
| good | 2 | \( 1 - 21.2T + 128T^{2} \) |
| 5 | \( 1 - 386.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 166.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 4.09e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.18e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 6.00e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.62e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 3.15e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 646.T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.14e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.90e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.13e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.14e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 1.01e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 9.99e5T + 1.17e12T^{2} \) |
| 61 | \( 1 - 2.14e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.67e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 1.57e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 3.71e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 5.47e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 7.86e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.83e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 5.62e5T + 8.07e13T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.13234045566789743017673249642, −8.753767142081002938941075875664, −7.57567269140811436436397576311, −6.24885494580598670214876265576, −6.06100313646674192284803975090, −5.09854023261302135305154268313, −4.21232750180491605477489120553, −3.01504373421481699226895322356, −2.23372422426676125493668351538, −1.27336063236825808287940917893,
1.27336063236825808287940917893, 2.23372422426676125493668351538, 3.01504373421481699226895322356, 4.21232750180491605477489120553, 5.09854023261302135305154268313, 6.06100313646674192284803975090, 6.24885494580598670214876265576, 7.57567269140811436436397576311, 8.753767142081002938941075875664, 10.13234045566789743017673249642
